Abstract:
We say n∈ N is perfect if σ(n)=2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n=pα∏j=1kqj2βj, where p, q1, ..., qk are distinct primes and p≡α≡1 (mod 4) . We prove that if βj≡1 (mod 3) or βj≡2 (mod 5) for all j, 1≤ j≤ k, then 3 n. We also prove as our main result that Ω(n)≥37, where Ω(n)=α+2∑j=1kβj. This improves a result of Sayers (Ω(n)≥ 29) given in 1986.